# Modeling complex systems by simple mathematical models using by yaofenji

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```									Representing complex engineering systems by
simple mathematical models

May 23, 2007
Jason Mayes
Outline

   Motivation
   Mathematical modeling
   Complex systems
   Objectives
   Present Work
   Using self-similarity
   Using fractional-order system identification
   Future Work
Mathematical Modeling
   A mathematical model is a mathematical description of a
physical system or process
   Design
   Prediction
   Control
   Mathematical modeling: finding a compromise between
an intractable problem and a model that sufficiently
describes the system
   Characteristics of a desirable model
   Simple
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Mathematical Modeling
x(t)
• Electro-mechanical model
• Equations of motion
• Experiments

Random Experimental Data

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The philosophy of analysis
   Many ways to find a mathematical model
Methodological Reductionism
   Descartes‟ 1637 Discourse on Method
   The world is like a machine…
   Everything can be reduced to many smaller, simpler things
   The best way to understand a system is to first gain a clear
understanding of its smallest subsystems
   Models are based on first principles
   Problems:
   Size and complexity
Holism
   Behavior must be studied on the level of the system as a whole

   Aristotle‟s Metaphysics –
“the whole is more than the sum of its parts”
Random Experimental Data

   Examples                                 1.2

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 Neural nets
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 On/off, PID, fuzzy logic
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 Expert systems
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   Black-box analysis is useful for describing or controlling
systems, but it doesn‟t explain observed behavior
A model-based compromise
   Mix of holistic and reductionist approaches
   Model-based parameter identification
   Can form a model containing a few free parameters
   Very common in heat transfer                N
   Nusselt number correlations, convection        F  N
   Contact resistance, fouling coefficients
   Heat exchangers
   Friction factor (Moody diagram)                  F

   Trend in science: HolisticReductionist analysis
   Example: chemical reactions
Complex systems
   Q: What is a complex system?
A: ???
   Properties:
   Size/complexity
   Non-linear
   Emergent phenomena
   Memory

   Our working definition: complex engineering systems
   Physically: any system composed of a large number of components
and interactions that creates difficulties in both understanding and
modeling
   Mathematically: a large system of coupled equations which are
either too complex or too large to admit a sufficiently useful
solution
Objectives
   To find simple mathematical models of complex
engineering systems
   Using both reductionist and holistic approaches to modeling

   Systems with similarity
   Both physical and mathematical similarity
   Can reduce complex mathematical models to simpler models

   Using fractional-order system identification
   Modeling complex systems as a black-box
   Better results than traditional integer-order models
Mathematical self-similarity
   Current Work: using a reductionist approach and taking advantage of
mathematical similarity to simplify complex models

   Mathematical similarity:
   Equations of the same form
   Repeating patterns or coupling in equations

   Reducible mathematical models
   Complex system: large system of ODEs
   System of first order ODEs  scalar ODE
   High-order scalar ODE  lower order scalar ODE
Mathematical Similarity
Reducing infinite-order ODEs

   High-order ODEs can be reduced in the Laplace domain

Geometric Series!!
Mathematical Similarity
Reducing infinite-order ODEs

   Can reduce an infinite-order (very large) ODE to a simple,
finite-order ODE

   Only useful in complex mechanical systems if infinite-order
ODEs occur in modeling
Mathematical Similarity
Reducing infinite sets of ODEs

   High-order (infinite) ODEs result
from large (infinite) equations sets
Mathematical Similarity
Reducing infinite sets of ODEs

   Consider a very large system of springs and masses
Mathematical Similarity
Reducing PDEs

   Infinite sets of ODEs also result from the reduction of PDEs
   Finite-volume
   Spectral methods
   Finite difference

   Example: heat equation
Mathematical Similarity
Reducing PDEs

   Finite-volume formulation
Mathematical Similarity
Reducing PDEs

   An infinite set of differential equations
Mathematical Similarity
Reducing PDEs

   Can now reduce the continued fraction

   Taking the limit

   Now take the inverse Laplace transform
Mathematical Similarity
Reducing PDEs

   Reduced the PDE in a way that we have extracted the heat
flux at the boundary

   Reduction process:

PDE  System of ODEs  Single high-order ODE  Single low-order ODE

   Alternative:
   Solve PDE
   Get a „global‟ solution
   Differentiate at the boundary
Mathematical Similarity
Applications of PDE reduction

   Only need a „local‟ solution
   Change of boundary conditions
   Blast furnace monitoring [Oldham and Spanier, 1974]
   Laser or cryogenic surgery
   Heat equation [Taler, 1996]
   Need only one thermocouple
Physical self-similarity

   Current work: using physical self-
similarity to reduce complex models

   Potential driven flows through
bifurcating trees

   Self-similar equations sets can also
result from physically self-similar
systems
Physical Similarity
A self-similar model

   The bifurcating tree geometry
   Geometry seen in a wide variety of
applications [Bejan, 2000]
   Potential-driven flow or transfer
   ex. heat, fluid, energy, etc.
   Conservation at bifurcation points

q1

q

q = q1+q2         q2
Physical Similarity
A self-similar model

   Assumptions
   Conservation at nodes
   Transfer governed by a linear operator
   i.e., for each branch:

   Large system of DAE‟s
   2n+1 -2 differential „branch‟ equations
   2n-1 continuity equations
Physical Similarity
Reduction

   System of DAEs can be reduced as before
   Regular coupling from physical similarity allows for reduction
   For N=1 generation network:

}
Physical Similarity
Reduction

   For N-generation network
Physical Similarity

   Similarity in operators can be used to further simplify
   Two forms of similarity:
   Similarity „within‟ a generation
   Symmetric networks: the operators within a generation are identical
   Asymmetric networks: the operators within a generation are not identical
   Similarity „between‟ generations
   Generation dependent operators depend on the generation in which the operator
occurs and change between successive generations
   Generation independent operators do not change between generations
   Four possible combinations
   Symmetric with generation independent operators
   Asymmetric with generation independent operators
   Symmetric with generation dependent operators
   Asymmetric with generation dependent operators
Physical Similarity
Symmetric and generation independent

   Can further reduce the continued fraction

For infinite networks:
Physical Similarity
Asymmetric and generation independent

   Further reduction:

For infinite networks:
Physical Similarity
Symmetric and generation dependent

   Further reduction:

For infinite networks:
• Can further reduce for
certain forms of
Physical Similarity
Asymmetric and generation dependent

   Further reduction:
   No general reduction
   Can reduce further for specific cases

For infinite networks:
Physical Similarity
Applications

   Viscoelasticity
    Asymmetric, generation independent
    Fractional-order viscoelastic models
    Springs (k) and dampers (µ)
    [Heymans and Bauwens, 1994]
Physical Similarity
Applications

   Laminar flow through bifurcating trees
    Symmetric, generation dependent
    Using laminar pipe flow model for
each branch:
Using Similarity
Summary of current work

   Representing complex engineering systems by
simple mathematical models

   From a reductionist perspective:
   Mathematical systems with similarity
   PDEs  Systems of ODEs  High-order scaler ODE  Lower-order scaler ODE

   Physical systems with similarity
   Use physical structure to reduce model
   Additional forms of similarity can offer further reduction
Fractional-order system ID
   Current work: using fractional-order models to better fit nearly
exponential experimental data
   System identification: building a dynamic mathematical model of a
system using measured/observed data
   Holistic approach to modeling
   Grey- or black-box modeling
   Typically assumes integer-order models
   Fractional-order models can
often do a better job describing
real physical systems
[Podlubny, 1999]

?
Fractional-order system ID
Nearly exponential transitions

   Focus on nearly exponential transitions
   A transition from one steady-state to
another, usually the result of step change
in input
   Typically assumed to be some combination
of exponentials
   Commonly modeled as first or second
order systems:

   Fractional-order models can often do a better job
Fractional-order system ID
Linear systems: a toy problem

     Consider the system

     System representation:

U (s)             6           Y (s)
s  6s  11s  6
3    2

     System identification:
Fractional-order system ID
Linear systems: a fractance device

   Fractance device – an electrical circuit having properties which lie
between resistance and capacitance or resistance and inductance.

   Has an impedance Z  (i )

   Construction: infinite bifurcating network of resistors and capacitors
[Nakagawa and Sorimachi, 1994]

   N = 1:

   N = ∞:
Fractional-order system ID
Linear systems: a fractance device

   For N=1, 3, 6, and 9 generation fractance devices

9
N=1
Fractional-order model is a special
case of the integer-order model
Fractional-order system ID
Nonlinear systems: a toy problem

     Consider the system

     System representation:

U (s)                      Y (s)

     System identification:
Fractional-order system ID
Nonlinear systems: experimental results

   Shell-and-tube heat exchanger
   Interested in a dynamic correlation
for the response to a step input
   Complex systems: reductionist approach
  System of PDEs
  Turbulence, recirculation
  Very detailed
   Holistic or black-box: experimentally determine response
  Measured hot-side outlet temperature

   Step change in cold-side inlet flow rate
Fractional-order system ID
Nonlinear systems: experimental results

   Experimental results:
Fractional-order system ID
Summary of current work

   Representing complex engineering systems by simple
mathematical models
   From a holistic perspective:
   Fractional-order models often better descriptors of complex
   Integer-order models are special cases of fractional-order
models
   Useful for modeling linear and non-linear systems
   Heat exchanger shown to be modeled well using a
fractional-order model [Aoki, 2006]
Future Work
   Work to proceed in two directions
   Reductionism
   Finding simple mathematical models of complex engineering
systems that exhibit similarity properties
   Holism
   Using simple fractional-order models
Future Work
Reduction using similarity

   Reducing PDEs
   Approximate boundary condition
transformations
   Pennes equation

   2-D PDEs
Future Work
Reduction using similarity

   Reducing physical systems
   Grid-like transfer networks
Future Work
Reduction using fractional-order models

   Using fractional-order models
   Continue heat exchanger experiments to better develop
dynamic correlation
   Fractional-order control:

   Fractional-order systems can be controlled more efficiently using
fractional-order control techniques
   Using fractional-order system identification and experimental
results to choose proper controller gains
   Similar to Ziegler-Nichols method of parameter tuning
   Based on offline time response characteristics
Questions?

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